Algebra - Factorization of polynomials (Remainder and Factor Theorems)

Definition of a Polynomial: A polynomial $f(x)$ in one variable $x$ is an algebraic expression of the form $a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$. Visually, the graph of a polynomial is a smooth, continuous curve without any breaks or sharp corners, where the degree of the polynomial determines the maximum number of times the curve can turn.

The Division Algorithm: For any polynomial $f(x)$ and a divisor $g(x)$, there exist unique polynomials $q(x)$ (quotient) and $r(x)$ (remainder) such that $f(x) = g(x) \cdot q(x) + r(x)$, where the degree of $r(x)$ is always less than the degree of $g(x)$.

The Remainder Theorem: If a polynomial $f(x)$ is divided by $(x - a)$, the remainder is equal to $f(a)$. Visually, this remainder represents the $y$-coordinate of the point on the graph of $y = f(x)$ where the $x$-coordinate is $a$. If the remainder is positive, the point is above the $x$-axis; if negative, it is below.

The Factor Theorem: A polynomial $(x - a)$ is a factor of $f(x)$ if and only if $f(a) = 0$. In graphical terms, if $(x - a)$ is a factor, the curve $y = f(x)$ will cross or touch the $x$-axis at the point $(a, 0)$, making $x = a$ an $x$-intercept of the graph.

General Remainder Rule: When a polynomial $f(x)$ is divided by $(ax - b)$, the remainder is $f(\frac{b}{a})$. If divided by $(ax + b)$, the remainder is $f(-\frac{b}{a})$. This is found by setting the linear divisor to zero ($ax - b = 0$) and solving for $x$.

Complete Factorization: To factorize a cubic polynomial $ax^3 + bx^2 + cx + d$, we first find one factor $(x - k)$ using the Factor Theorem (by testing factors of the constant term $d$). We then use synthetic division or long division to divide the polynomial by $(x - k)$ to obtain a quadratic quotient, which is then factorized further into two linear factors if possible.

Multiple Factors and Simultaneous Equations: If a polynomial has two unknown constants (e.g., $a$ and $b$) and we are given two factors or two remainders, we can substitute the values into $f(x)$ to form a system of two linear equations. Solving these equations simultaneously reveals the values of $a$ and $b$.